Applied Probability

ISSN 1050-5164 (print) ISSN 2168-8737 (online) THE ANNALS of APPLIED PROBABILITY

AN OFFICIAL JOURNAL OF THE INSTITUTE OF MATHEMATICAL STATISTICS

Articles Social contact processes and the partner model ERIC FOXALL,RODERICK EDWARDS AND P. VA N D E N DRIESSCHE 1297 Diverse market models of competing Brownian particles with splits andmergers...... IOANNIS KARATZAS AND ANDREY SARANTSEV 1329 A positive temperature phase transition in random hypergraph 2-coloring VICTOR BAPST,AMIN COJA-OGHLAN AND FELICIA RASSMANN 1362 Propagation of chaos for interacting particles subject to environmental noise MICHELE COGHI AND FRANCO FLANDOLI 1407 Approximations of stochastic partial differential equations GIULIA DI NUNNO AND TUSHENG ZHANG 1443 Local asymptotics for controlled martingales SCOTT N. ARMSTRONG AND OFER ZEITOUNI 1467 Stein estimation of the intensity of a spatial homogeneous Poisson point process MARIANNE CLAUSEL,JEAN-FRANÇOIS COEURJOLLY AND JÉRÔME LELONG 1495 A probabilistic approach to mean ﬁeld games with major and minor players RENÉ CARMONA AND XIUNENG ZHU 1535 Estimation for stochastic damping Hamiltonian systems under partial observation. III. Diffusionterm.... PATRICK CATTIAUX,JOSÉ R. LEÓN AND CLÉMENTINE PRIEUR 1581 From transience to recurrence with Poisson tree frogs CHRISTOPHER HOFFMAN,TOBIAS JOHNSON AND MATTHEW JUNGE 1620 Bernoulli and tail-dependence compatibility PAUL EMBRECHTS,MARIUS HOFERT AND RUODU WANG 1636 Beyond universality in random matrix theory ALAN EDELMAN,A.GUIONNET AND S. PÉCHÉ 1659 Super-replication with nonlinear transaction costs and volatility uncertainty PETER BANK,YAN DOLINSKY AND SELIM GÖKAY 1698 ThesnappingoutBrownianmotion...... ANTOINE LEJAY 1727 Backward stochastic differential equation driven by a marked point process: An elementary approach with an application to optimal control FULVIA CONFORTOLA,MARCO FUHRMAN AND JEAN JACOD 1743 Entropic Ricci curvature bounds for discrete interacting systems MAX FATHI AND JAN MAAS 1774 Hack’s law in a drainage network model: A Brownian web approach RAHUL ROY,KUMARJIT SAHA AND ANISH SARKAR 1807 Gaussian ﬂuctuations for linear spectral statistics of large random covariancematrices...... JAMAL NAJIM AND JIANFENG YAO 1837 Duality theory for portfolio optimisation under transaction costs CHRISTOPH CZICHOWSKY AND WALTER SCHACHERMAYER 1888 A note on the expansion of the smallest eigenvalue distribution of the LUE atthehardedge...... FOLKMAR BORNEMANN 1942

Vol. 26, No. 3—June 2016 The Annals of Applied Probability 2016, Vol. 26, No. 3, 1297–1328 DOI: 10.1214/15-AAP1117 © Institute of Mathematical Statistics, 2016

SOCIAL CONTACT PROCESSES AND THE PARTNER MODEL

BY ERIC FOXALL1,RODERICK EDWARDS2 AND P. VA N D E N DRIESSCHE2 University of Victoria

We consider a stochastic model of infection spread on the complete graph on N vertices incorporating dynamic partnerships, which we assume to be monogamous. This can be seen as a variation on the contact process in which some form of edge dynamics determines the set of contacts at each moment in time. We identify a basic reproduction number R0 with the property that if R0 < 1 the infection dies out by time O(log N), while if R0 > 1 the infection survives for an amount of time eγN for some γ>0 and hovers around a uniquely determined metastable proportion of infectious individuals. The proof in both cases relies on comparison to a set of mean-ﬁeld equations when the infection is widespread, and to a branching process when the infection is sparse.

REFERENCES

[1] BEZUIDENHOUT,C.andGRIMMETT, G. (1990). The critical contact process dies out. Ann. Probab. 18 1462–1482. MR1071804 [2] BROMAN, E. I. (2007). Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment. Ann. Probab. 35 2263–2293. MR2353388 [3] CHATTERJEE,S.andDURRETT, R. (2009). Contact processes on random graphs with power law degree distributions have critical value 0. Ann. Probab. 37 2332–2356. MR2573560 [4] DURRETT, R. (1980). On the growth of one-dimensional contact processes. Ann. Probab. 8 890–907. MR0586774 [5] DURRETT, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cam- bridge. MR2722836 [6] HARRIS, T. E. (1978). Additive set-valued Markov processes and graphical methods. Ann. Probab. 6 355–378. MR0488377 [7] HARRIS, T. E. (2002). The Theory of Branching Processes. Dover, Mineola, NY. MR1991122 [8] LIGGETT, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York. MR0776231 [9] LIGGETT, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Pro- cesses. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 324. Springer, Berlin. MR1717346 [10] MOUNTFORD,T.,VALESIN,D.andYAO, Q. (2013). Metastable densities for the contact process on power law random graphs. Electron. J. Probab. 18 1–36. MR3145050 [11] PEMANTLE, R. (1992). The contact process on trees. Ann. Probab. 20 2089–2116. MR1188054

MSC2010 subject classiﬁcations. Primary 60J25; secondary 92B99. Key words and phrases. SIS model, contact process, interacting particle systems. [12] PETERSON, J. (2011). The contact process on the complete graph with random vertex- dependent infection rates. Stochastic Process. Appl. 121 609–629. MR2763098 [13] REMENIK, D. (2008). The contact process in a dynamic random environment. Ann. Appl. Probab. 18 2392–2420. MR2474541 [14] VA N D E N DRIESSCHE,P.andWATMOUGH, J. (2002). Reproduction numbers and sub- threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180 29–48. MR1950747 The Annals of Applied Probability 2016, Vol. 26, No. 3, 1329–1361 DOI: 10.1214/15-AAP1118 © Institute of Mathematical Statistics, 2016

DIVERSE MARKET MODELS OF COMPETING BROWNIAN PARTICLES WITH SPLITS AND MERGERS1

BY IOANNIS KARATZAS AND ANDREY SARANTSEV Columbia University and University of Washington

We study models of regulatory breakup, in the spirit of Strong and Fouque [Ann. Finance 7 (2011) 349–374] but with a fluctuating number of companies. An important class of market models is based on systems of competing Brownian particles: each company has a capitalization whose loga- rithm behaves as a Brownian motion with drift and diffusion coefficients de- pending on its current rank. We study such models with a fluctuating number of companies: If at some moment the share of the total market capitalization of a company reaches a fixed level, then the company is split into two parts of random size. Companies are also allowed to merge, when an exponential clock rings. We find conditions under which this system is nonexplo- sive (i.e., the number of companies remains finite at all times) and diverse, yet does not admit arbitrage opportunities.

REFERENCES

[1] AUDRINO,F.,FERNHOLZ,E.R.andFERRETTI, R. G. (2007). A forecasting model for stock market diversity. Ann. Finance 3 213–240. [2] BANNER,A.D.,FERNHOLZ,E.R.andKARATZAS, I. (2005). Atlas models of equity markets. Ann. Appl. Probab. 15 2296–2330. MR2187296 [3] BANNER,A.D.andGHOMRASNI, R. (2008). Local times of ranked continuous semimartingales. Stochastic Process. Appl. 118 1244–1253. MR2428716 [4] BASS,R.F.andPARDOUX, É. (1987). Uniqueness for diffusions with piecewise constant coefﬁcients. Probab. Theory Related Fields 76 557–572. MR0917679 [5] BORODIN,A.N.andSALMINEN, P. (2002). Handbook of Brownian Motion—Facts and For- mulae, 2nd ed. Birkhäuser, Basel. MR1912205 [6] CHATTERJEE,S.andPAL, S. (2010). A phase transition behavior for Brownian motions interacting through their ranks. Probab. Theory Related Fields 147 123–159. MR2594349 [7] DEMBO,A.andZEITOUNI, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin. MR2571413 [8] FERNHOLZ, E. R. (2002). Stochastic Portfolio Theory. Applications of Mathematics (New York) 48. Springer, New York. MR1894767 [9] FERNHOLZ,E.R.,ICHIBA,T.andKARATZAS, I. (2013). Two Brownian particles with rank- based characteristics and skew-elastic collisions. Stochastic Process. Appl. 123 2999– 3026. MR3062434 [10] FERNHOLZ,E.R.,ICHIBA,T.andKARATZAS, I. (2013). A second-order stock market model. Ann. Finance 9 439–454. MR3082660

MSC2010 subject classifications. Primary 60K35, 60J60; secondary 91B26. Key words and phrases. Competing Brownian particles, splits, mergers, diverse markets, arbitrage opportunity, portfolio. [11] FERNHOLZ,E.R.andKARATZAS, I. (2005). Relative arbitrage in volatility-stabilized markets. Ann. Finance 1 149–177. [12] FERNHOLZ,E.R.andKARATZAS, I. (2009). Stochastic portfolio Theory: An Overview. Handb. Numer. Anal. 15 89–167. [13] FERNHOLZ,E.R.,KARATZAS,I.andKARDARAS, C. (2005). Diversity and relative arbitrage in equity markets. Finance Stoch. 9 1–27. MR2210925 [14] ICHIBA, T. (2009). Topics in multi-dimensional diffusion theory: Attainability, reflection, ergodicity and rankings. Ph.D. thesis, Columbia University, ProQuest LLC, Ann Arbor, MI. MR2717746 [15] ICHIBA,T.andKARATZAS, I. (2010). On collisions of Brownian particles. Ann. Appl. Probab. 20 951–977. MR2680554 [16] ICHIBA,T.,KARATZAS,I.andSHKOLNIKOV, M. (2013). Strong solutions of stochastic equations with rank-based coefficients. Probab. Theory Related Fields 156 229–248. MR3055258 [17] ICHIBA,T.,PAL,S.andSHKOLNIKOV, M. (2013). Convergence rates for rank-based models with applications to portfolio theory. Probab. Theory Related Fields 156 415–448. MR3055264 [18] ICHIBA,T.,PAPATHANAKOS,V.,BANNER,A.,KARATZAS,I.andFERNHOLZ, E. R. (2011). Hybrid atlas models. Ann. Appl. Probab. 21 609–644. MR2807968 [19] IKEDA,N.andWATANABE, S. (1989). Stochastic Differential Equations and Diffusion Pro- cesses, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam. MR1011252 [20] KARATZAS,I.,PAL,S.andSHKOLNIKOV, M. (2016). Systems of Brownian particles with asymmetric collisions. Ann. Inst. Henri Poincaré Probab. Stat. 52 323–354. [21] KARATZAS,I.andSHREVE, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York. MR1121940 [22] KARDARAS, C. (2008). Balance, growth and diversity of financial markets. Ann. Finance 4 369–397. [23] KAZAMAKI, N. (1994). Continuous Exponential Martingales and BMO. Lecture Notes in Math. 1579. Springer, Berlin. MR1299529 [24] OSTERRIEDER,J.R.andRHEINLANDER, T. (2006). Arbitrage opportunities in diverse markets via a non-equivalent measure change. Ann. Finance 2 287–301. [25] PAL,S.andPITMAN, J. (2008). One-dimensional Brownian particle systems with rank- dependent drifts. Ann. Appl. Probab. 18 2179–2207. MR2473654 [26] PAL,S.andSHKOLNIKOV, M. (2014). Concentration of measure for Brownian particle systems interacting through their ranks. Ann. Appl. Probab. 24 1482–1508. MR3211002 [27] RUF,J.andRUNGGALDIER, W. (2013). A systematic approach to constructing market models with arbitrage. Preprint. Available at arXiv:1309.1988. [28] SARANTSEV, A. (2014). On a class of diverse market models. Ann. Finance 10 291–314. MR3193785 [29] SARANTSEV, A. (2015). Infinite systems of competing Brownian particles. Preprint. Available at arXiv:1403.4229. [30] SARANTSEV, A. (2015). Multiple collisions in systems of competing Brownian particles. Preprint. Available at arXiv:1309.2621. [31] SARANTSEV, A. (2015). Triple and simultaneous collisions of competing Brownian particles. Electron. J. Probab. 20 1–28. MR3325099 [32] SHKOLNIKOV, M. (2011). Competing particle systems evolving by interacting Lévy processes. Ann. Appl. Probab. 21 1911–1932. MR2884054 [33] STRONG, W. (2014). Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension. Finance Stoch. 18 487–514. MR3232014 [34] STRONG,W.andFOUQUE, J.-P. (2011). Diversity and arbitrage in a regulatory breakup model. Ann. Finance 7 349–374. MR2879921 The Annals of Applied Probability 2016, Vol. 26, No. 3, 1362–1406 DOI: 10.1214/15-AAP1119 © Institute of Mathematical Statistics, 2016

A POSITIVE TEMPERATURE PHASE TRANSITION IN RANDOM HYPERGRAPH 2-COLORING1

BY VICTOR BAPST,AMIN COJA-OGHLAN AND FELICIA RASSMANN Goethe University

Diluted mean-ﬁeld models are graphical models in which the geometry of interactions is determined by a sparse random graph or hypergraph. Based on a nonrigorous but analytic approach called the “cavity method”, physicists have predicted that in many diluted mean-ﬁeld models a phase transition oc- curs as the inverse temperature grows from 0 to ∞ [Proc. National Academy of Sciences 104 (2007) 10318–10323]. In this paper, we establish the existence and asymptotic location of this so-called condensation phase transition in the random hypergraph 2-coloring problem.

REFERENCES

[1] ACHLIOPTAS,D.andCOJA-OGHLAN, A. (2008). Algorithmic barriers from phase transitions. Proc.49th FOCS 793–802. [2] ACHLIOPTAS,D.andMOORE, C. (2002). On the 2-colorability of random hypergraphs. In Randomization and Approximation Techniques in Computer Science. Lecture Notes in Computer Science 2483 78–90. Springer, Berlin. MR2047021 [3] ACHLIOPTAS,D.andTHEODOROPOULOS, P. (2014). Manuscript in preparation. [4] BAPST,V.,COJA-OGHLAN,A.,HETTERICH,S.,RASSMANN,F.andVILENCHIK, D. (2014). The condensation phase transition in random graph coloring. Preprint. Available at arXiv:1404.5513. [5] BAYATI,M.,GAMARNIK,D.andTETALI, P. (2013). Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. Ann. Probab. 41 4080–4115. [6] COJA-OGHLAN,A.andZDEBOROVÁ, L. (2012). The condensation transition in random hypergraph 2-coloring. In Proceedings of the Twenty-Third Annual ACM–SIAM Symposium on Discrete Algorithms 241–250. ACM, New York. MR3205212 [7] CONTUCCI,P.,DOMMERS,S.,GIARDINÀ,C.andSTARR, S. (2013). Antiferromagnetic Potts model on the Erdos–Rényi˝ random graph. Comm. Math. Phys. 323 517–554. MR3096530 [8] DALL’ASTA,L.,RAMEZANPOUR,A.andZECCHINA, R. (2008). Entropy landscape and non- Gibbs solutions in constraint satisfaction problems. Phys. Rev. E 77 031118. [9] JANSON,S.,ŁUCZAK,T.andRUCINSKI´ , A. (2000). Random Graphs. Wiley, New York. [10] KAUZMANN, W. (1948). The nature of the glassy state and the behavior of liquids at low temperatures. Chem. Rev. 43 219–256. [11] KRZAKALA,F.,MÉZARD,M.,SAUSSET,F.,SUN,Y.andZDEBOROVÁ, L. (2012). Proba- bilistic reconstruction in compressed sensing: Algorithms, phase diagrams, and threshold achieving matrices. J. Stat. Mech. P08009. [12] KRZAKALA,F.,MONTANARI,A.,RICCI-TERSENGHI,F.,SEMERJIAN,G.andZDE- BOROVA, L. (2007). Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. National Academy of Sciences 104 10318–10323.

MSC2010 subject classiﬁcations. 05C80, 68Q87, 05C15. Key words and phrases. Discrete structures, random hypergraphs, phase transitions, positive temperature. [13] KRZAKALA,F.andZDEBOROVÁ, L. (2008). Potts glass on random graphs. Europhys. Lett. 81 57005. [14] MCDIARMID, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics 195–248. Springer, Berlin. [15] MÉZARD,M.andMONTANARI, A. (2009). Information, Physics and Computation. Oxford Univ. Press, London. The Annals of Applied Probability 2016, Vol. 26, No. 3, 1407–1442 DOI: 10.1214/15-AAP1120 © Institute of Mathematical Statistics, 2016

PROPAGATION OF CHAOS FOR INTERACTING PARTICLES SUBJECT TO ENVIRONMENTAL NOISE

BY MICHELE COGHI AND FRANCO FLANDOLI Scuola Normale Superiore and University of Pisa

A system of interacting particles described by stochastic differential equations is considered. As oppopsed to the usual model, where the noise perturbations acting on different particles are independent, here the particles are subject to the same space-dependent noise, similar to the (noninteracting) particles of the theory of diffusion of passive scalars. We prove a result of propagation of chaos and show that the limit PDE is stochastic and of invis- cid type, as opposed to the case when independent noises drive the different particles.

REFERENCES

[1] AMBROSIO,L.,GIGLI,N.andSAVA R É , G. (2008). Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed. Birkhäuser, Basel. MR2401600 [2] CARLEN,E.A.,CARVALHO,M.C.,LE ROUX,J.,LOSS,M.andVILLANI, C. (2010). En- tropy and chaos in the Kac model. Kinet. Relat. Models 3 85–122. MR2580955 [3] DELARUE,F.,FLANDOLI,F.andVINCENZI, D. (2014). Noise prevents collapse of Vlasov– Poisson point charges. Comm. Pure Appl. Math. 67 1700–1736. MR3251910 [4] DERMOUNE, A. (2003). Propagation and conditional propagation of chaos for pressureless gas equations. Probab. Theory Related Fields 126 459–476. MR2001194 [5] DOBRUŠIN, R. L. (1979). Vlasov equations. Funktsional. Anal. i Prilozhen. 13 48–58, 96. MR0541637 [6] FOURNIER,N.andGUILLIN, A. (2013). On the rate of convergence in wasserstein distance of the empirical measure. Available at arXiv:1312.2128v1. [7] FOURNIER,N.,HAURAY,M.andMISCHLER, S. (2014). Propagation of chaos for the 2D viscous vortex model. J. Eur. Math. Soc.(JEMS) 16 1423–1466. MR3254330 [8] HAURAY,M.andMISCHLER, S. (2014). On Kac’s chaos and related problems. J. Funct. Anal. 266 6055–6157. MR3188710 [9] KUNITA, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Stud- ies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge. MR1070361 [10] PILIPENKO, A. Y. (2005). Measure-valued diffusions and continuous systems of interacting particles in a random environment. Ukraïn. Mat. Zh. 57 1289–1301. MR2216047 [11] PILIPENKO, A. Y. (2006). A functional central limit theorem for ﬂows generated by stochastic equations with interaction. Nel¯ın¯ı˘ın¯ıKoliv. 9 85–97. MR2369777 [12] REVUZ,D.andYOR, M. (1999). Continuous Martingales and Brownian Motion,3rded. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin. MR1725357 [13] SKOROKHOD, A. V. (1997). Measure-valued diffusion. Ukraïn. Mat. Zh. 49 458–464. MR1472230

MSC2010 subject classiﬁcations. 60K35, 82C22, 60K37, 60H15. Key words and phrases. Interacting particle system, propagation of chaos, mean ﬁeld limit, Kraichnan noise, Wasserstain metric. [14] SZNITMAN, A.-S. (1991). Topics in propagation of chaos. In École D’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math. 1464 165–251. Springer, Berlin. MR1108185 [15] ZHENG, W. A. (1995). Conditional propagation of chaos and a class of quasilinear PDE’s. Ann. Probab. 23 1389–1413. MR1349177 The Annals of Applied Probability 2016, Vol. 26, No. 3, 1443–1466 DOI: 10.1214/15-AAP1122 © Institute of Mathematical Statistics, 2016

APPROXIMATIONS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

BY GIULIA DI NUNNO AND TUSHENG ZHANG University of Oslo and University of Manchester

In this paper, we show that solutions of stochastic partial differential equations driven by Brownian motion can be approximated by stochastic partial differential equations forced by pure jump noise/random kicks. Applica- tions to stochastic Burgers equations are discussed.

REFERENCES

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MSC2010 subject classiﬁcations. Primary 60H15; secondary 93E20, 35R60. Key words and phrases. Stochastic partial differential equations, approximations, jump noise, tightness, weak convergence, stochastic Burgers equations. [14] GYÖNGY,I.andKRYLOV, N. V. (1981/82). On stochastics equations with respect to semimartingales. II. Itô formula in Banach spaces. Stochastics 6 153–173. MR0665398 [15] IKEDA,N.andWATANABE, S. (1989). Stochastic Differential Equations and Diffusion Pro- cesses, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam. MR1011252 [16] JAKUBOWSKI, A. (1986). On the Skorokhod topology. Ann. Inst. Henri Poincaré Probab. Stat. 22 263–285. MR0871083 [17] KALLENBERG, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York. MR1876169 [18] KRYLOV,N.V.andROZOVSKII˘, B. L. (1979). Stochastic evolution equations. In Current Problems in Mathematics, Vol.14(Russian) 71–147. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow. MR0570795 [19] LIU,W.andRÖCKNER, M. (2013). Local and global well-posedness of SPDE with generalized coercivity conditions. J. Differential Equations 254 725–755. MR2990049 [20] PARDOUX, E. (1979). Stochastic partial differential equations and ﬁltering of diffusion processes. Stochastics 3 127–167. MR0553909 [21] PRÉVÔT,C.andRÖCKNER, M. (2007). A Concise Course on Stochastic Partial Differential Equations. Springer, Berlin. MR2329435 [22] RÖCKNER,M.andZHANG, T. (2007). Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles. Potential Anal. 26 255–279. MR2286037 [23] TRIEBEL, H. (1978). Interpolation Theory, Function Spaces, Differential Operators.North- Holland, Amsterdam. MR0503903 The Annals of Applied Probability 2016, Vol. 26, No. 3, 1467–1494 DOI: 10.1214/15-AAP1123 © Institute of Mathematical Statistics, 2016

LOCAL ASYMPTOTICS FOR CONTROLLED MARTINGALES

∗ BY SCOTT N. ARMSTRONG AND OFER ZEITOUNI1,†,‡ Université Paris-Dauphine∗, Weizmann Institute of Science† and New York University‡

We consider controlled martingales with bounded steps where the controller is allowed at each step to choose the distribution of the next step, and where the goal is to hit a fixed ball at the origin at time n. We show that the algebraic rate of decay (as n increases to infinity) of the value function in the discrete setup coincides with its continuous counterpart, provided a reachability assumption is satisfied. We also study in some detail the uniformly elliptic case and obtain explicit bounds on the rate of decay. This generalizes and improves upon several recent studies of the one dimensional case, and is a discrete analogue of a stochastic control problem recently investigated in Armstrong and Trokhimtchouck [Calc. Var. Partial Differential Equations 38 (2010) 521– 540].

REFERENCES

[1] ALEXANDER, K. S. (2013). Controlled random walk with a target site. Electron. Commun. Probab. 18 1–6. MR3070909 [2] ARMSTRONG,S.N.andTROKHIMTCHOUK, M. (2010). Long-time asymptotics for fully nonlinear homogeneous parabolic equations. Calc. Var. Partial Differential Equations 38 521–540. MR2647131 [3] AZUMA, K. (1967). Weighted sums of certain dependent random variables. Tôhoku Math. J.(2) 19 357–367. MR0221571 [4] BERTSEKAS,D.P.andSHREVE, S. E. (1978). Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering 139. Academic Press, New York. MR0511544 [5] BUCKDAHN,R.,MA,J.andRAINER, C. (2008). Stochastic control problems for systems driven by normal martingales. Ann. Appl. Probab. 18 632–663. MR2399708 [6] CAFFARELLI,L.A.andCABRÉ, X. (1995). Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications 43. Amer. Math. Soc., Providence, RI. MR1351007 [7] DYNKIN,E.B.andYUSHKEVICH, A. A. (1979). Controlled Markov Processes. Grundlehren der Mathematischen Wissenschaften 235. Springer, Berlin. MR0554083 [8] GUREL-GUREVICH,O.,PERES,Y.andZEITOUNI, O. (2014). Localization for controlled random walks and martingales. Electron. Commun. Probab. 19 1–8. MR3208322 [9] HALL,P.andHEYDE, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York. MR0624435

MSC2010 subject classiﬁcations. 60G42, 93E20. Key words and phrases. Stochastic control, martingale, nonlinear parabolic equation. [10] KAMIN,S.,PELETIER,L.A.andVÁZQUEZ, J. L. (1991). On the Barenblatt equation of elastoplastic ﬁltration. Indiana Univ. Math. J. 40 1333–1362. MR1142718 [11] KRYLOV, N. V. (1980). Controlled Diffusion Processes. Applications of Mathematics 14. Springer, Berlin. MR0601776 [12] LEE,J.,PERES,Y.andSMART, C. K. (2015). A gaussian upper bound for martingale small ball probabilities. Preprint. Available at arXiv:1405.5980 [math.PR]. [13] MCNAMARA, J. M. (1985). A regularity condition on the transition probability measure of a diffusion process. Stochastics 15 161–182. MR0869198 [14] TRUDINGER, N. S. (1988). Hölder gradient estimates for fully nonlinear elliptic equations. Proc. Roy. Soc. Edinburgh Sect. A 108 57–65. MR0931007 The Annals of Applied Probability 2016, Vol. 26, No. 3, 1495–1534 DOI: 10.1214/15-AAP1124 © Institute of Mathematical Statistics, 2016

STEIN ESTIMATION OF THE INTENSITY OF A SPATIAL HOMOGENEOUS POISSON POINT PROCESS

BY MARIANNE CLAUSEL, JEAN-FRANÇOIS COEURJOLLY AND JÉRÔME LELONG University Grenoble Alpes

In this paper, we revisit the original ideas of Stein and propose an estimator of the intensity parameter of a homogeneous Poisson point process deﬁned on Rd and observed on a bounded window. The procedure is based on a new integration by parts formula for Poisson point processes. We show that our Stein estimator outperforms the maximum likelihood estimator in terms of mean squared error. In many practical situations, we obtain a gain larger than 30%.

REFERENCES

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MSC2010 subject classifications. Primary 60G55; secondary 60H07. Key words and phrases. Stein formula, Malliavin calculus, superefficient estimator, intensity estimation, spatial point process. JAMES,W.andSTEIN, C. (1961). Estimation with quadratic loss. In Proc.4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 361–379. Univ. California Press, Berkeley, CA. MR0133191 MØLLER,J.andWAAGEPETERSEN, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability 100. Chapman & Hall/CRC, Boca Raton, FL. MR2004226 MURR, R. (2012). Reciprocal classes of Markov processes. An approach with duality formulae. Ph.D. thesis. PRAT,J.-J.andPRIVAULT, N. (1999). Explicit stochastic analysis of Brownian motion and point measures on Riemannian manifolds. J. Funct. Anal. 167 201–242. MR1710625 PRIVAULT, N. (1994). Chaotic and variational calculus in discrete and continuous time for the Pois- son process. Stoch. Stoch. Rep. 51 83–109. MR1380764 PRIVAULT, N. (2009). Stochastic Analysis in Discrete and Continuous Settings with Normal Martin- gales. Lecture Notes in Math. 1982. Springer, Berlin. MR2531026 PRIVAULT,N.andRÉVEILLAC, A. (2006). Superefficient drift estimation on the Wiener space. C. R. Math. Acad. Sci. Paris 343 607–612. MR2269873 PRIVAULT,N.andRÉVEILLAC, A. (2008). Stein estimation for the drift of Gaussian processes using the Malliavin calculus. Ann. Statist. 36 2531–2550. MR2458197 PRIVAULT,N.andRÉVEILLAC, A. (2009). Stein estimation of Poisson process intensities. Stat. Inference Stoch. Process. 12 37–53. MR2486115 PRIVAULT,N.andTORRISI, G. L. (2011). Density estimation of functionals of spatial point processes with application to wireless networks. SIAM J. Math. Anal. 43 1311–1344. MR2821586 RÖCKNER,M.andSCHIED, A. (1999). Rademacher’s theorem on configuration spaces and applications. J. Funct. Anal. 169 325–356. MR1730565 RUBINSTEIN,R.Y.andSHAPIRO, A. (1993). Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method. Wiley, Chichester. MR1241645 STEIN, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probabil- ity, 1954–1955, Vol. I 197–206. Univ. California Press, Berkeley and Los Angeles. MR0084922 STEIN, C. M. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135–1151. MR0630098 STOYAN,D.,KENDALL,W.S.andMECKE, J. (1987). Stochastic Geometry and Its Applications. Wiley, Chichester. MR0895588 TRIEBEL, H. (1983). Theory of Functions Spaces. Birkhauser, Basel. ZUILY, C. (2002). Eléments de distributions et d’équations aux dérivées partielles. Dunod, Paris. The Annals of Applied Probability 2016, Vol. 26, No. 3, 1535–1580 DOI: 10.1214/15-AAP1125 © Institute of Mathematical Statistics, 2016

A PROBABILISTIC APPROACH TO MEAN FIELD GAMES WITH MAJOR AND MINOR PLAYERS

BY RENÉ CARMONA AND XIUNENG ZHU Princeton University

We propose a new approach to mean field games with major and minor players. Our formulation involves a two player game where the optimization of the representative minor player is standard while the major player faces an optimization over conditional McKean–Vlasov stochastic differential equations. The definition of this limiting game is justified by proving that its solution provides approximate Nash equilibriums for large finite player games. This proof depends upon the generalization of standard results on the propagation of chaos to conditional dynamics. Because it is of independent interest, we prove this generalization in full detail. Using a conditional form of the Pontryagin stochastic maximum principle (proven in the Appendix), we reduce the solution of the mean field game to a forward–backward system of stochastic differential equations of the conditional McKean–Vlasov type, which we solve in the linear quadratic setting. We use this class of models to show that Nash equilibriums in our formulation can be different from those originally found in the literature.

REFERENCES

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ESTIMATION FOR STOCHASTIC DAMPING HAMILTONIAN SYSTEMS UNDER PARTIAL OBSERVATION. III. DIFFUSION TERM

∗ BY PATRICK CATTIAUX ,JOSÉ R. LEÓN†,1,2 AND CLÉMENTINE PRIEUR‡,1 Université de Toulouse∗, Universidad Central de Venezuela† and Université Grenoble Alpes‡

This paper is the third part of our study started with Cattiaux, León and Prieur [Stochastic Process. Appl. 124 (2014) 1236–1260; ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 359–384]. For some ergodic Hamiltonian systems, we obtained a central limit theorem for a nonparametric estimator of the invariant density [Stochastic Process. Appl. 124 (2014) 1236–1260] and of the drift term [ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 359–384], under partial observation (only the positions are observed). Here, we obtain similarly a central limit theorem for a nonparametric estimator of the diffusion term.

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FROM TRANSIENCE TO RECURRENCE WITH POISSON TREE FROGS

∗ BY CHRISTOPHER HOFFMAN ,1,TOBIAS JOHNSON†,2 ∗ AND MATTHEW JUNGE University of Washington∗ and University of Southern California†

Consider the following interacting particle system on the d-ary tree, known as the frog model: Initially, one particle is awake at the root and i.i.d. Poisson many particles are sleeping at every other vertex. Particles that are awake perform simple random walks, awakening any sleeping particles they encounter. We prove that there is a phase transition between transience and recurrence as the initial density of particles increases, and we give the order of the transition up to a logarithmic factor.

REFERENCES

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BERNOULLI AND TAIL-DEPENDENCE COMPATIBILITY

∗ BY PAUL EMBRECHTS ,1,MARIUS HOFERT†,2 AND RUODU WANG†,2,3 ETH Zurich∗ and University of Waterloo†

The tail-dependence compatibility problem is introduced. It raises the question whether a given d × d-matrix of entries in the unit interval is the matrix of pairwise tail-dependence coefﬁcients of a d-dimensional random vector. The problem is studied together with Bernoulli-compatible matrices, that is, matrices which are expectations of outer products of random vectors with Bernoulli margins. We show that a square matrix with diagonal entries being 1 is a tail-dependence matrix if and only if it is a Bernoulli-compatible matrix multiplied by a constant. We introduce new copula models to construct tail-dependence matrices, including commonly used matrices in statistics.

REFERENCES

[1] BERMAN,A.andSHAKED-MONDERER, N. (2003). Completely Positive Matrices. World Sci- entiﬁc, River Edge, NJ. MR1986666 [2] BLUHM,C.andOVERBECK, L. (2007). Structured Credit Portfolio Analysis, Baskets & CDOs. Chapman & Hall/CRC, Boca Raton, FL. MR2260812 [3] BLUHM,C.,OVERBECK,L.andWAGNER, C. (2002). An Introduction to Credit Risk Model- ing. Chapman & Hall, London. [4] CHAGANTY,N.R.andJOE, H. (2006). Range of correlation matrices for dependent Bernoulli random variables. Biometrika 93 197–206. MR2277750 [5] DHAENE,J.andDENUIT, M. (1999). The safest dependence structure among risks. Insurance Math. Econom. 25 11–21. MR1718527 [6] EMBRECHTS,P.,MCNEIL,A.J.andSTRAUMANN, D. (2002). Correlation and dependence in risk management: Properties and pitfalls. In Risk Management: Value at Risk and Beyond (Cambridge, 1998) 176–223. Cambridge Univ. Press, Cambridge. MR1892190 [7] FIEBIG,U.,STROKORB,K.andSCHLATHER, M. (2014). The realization problem for tail correlation functions. Available at http://arxiv.org/abs/1405.6876. [8] JOE, H. (1997). Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73. Chapman & Hall, London. MR1462613 [9] JOE, H. (2015). Dependence Modeling with Copulas. Monographs on Statistics and Applied Probability 134. CRC Press, Boca Raton, FL. MR3328438 [10] KORTSCHAK,D.andALBRECHER, H. (2009). Asymptotic results for the sum of dependent non-identically distributed random variables. Methodol. Comput. Appl. Probab. 11 279– 306. MR2511246 [11] LIEBSCHER, E. (2008). Construction of asymmetric multivariate copulas. J. Multivariate Anal. 99 2234–2250. MR2463386 [12] MCNEIL,A.J.,FREY,R.andEMBRECHTS, P. (2005). Quantitative Risk Management: Con- cepts, Techniques and Tools. Princeton Univ. Press, Princeton, NJ. MR2175089

MSC2010 subject classiﬁcations. 60E05, 62H99, 62H20, 62E15, 62H86. Key words and phrases. Tail dependence, Bernoulli random vectors, compatibility, matrices, copulas, insurance application. [13] NIKOLOULOPOULOS,A.K.,JOE,H.andLI, H. (2009). Extreme value properties of multivariate t copulas. Extremes 12 129–148. MR2515644 [14] RÜSCHENDORF, L. (1981). Characterization of dependence concepts in normal distributions. Ann. Inst. Statist. Math. 33 347–359. MR0637748 [15] STROKORB,K.,BALLANI,F.andSCHLATHER, M. (2015). Tail correlation functions of max- stable processes: Construction principles, recovery and diversity of some mixing max- stable processes with identical TCF. Extremes 18 241–271. MR3351816 [16] YANG,J.,QI,Y.andWANG, R. (2009). A class of multivariate copulas with bivariate Fréchet marginal copulas. Insurance Math. Econom. 45 139–147. MR2549874 The Annals of Applied Probability 2016, Vol. 26, No. 3, 1659–1697 DOI: 10.1214/15-AAP1129 © Institute of Mathematical Statistics, 2016

BEYOND UNIVERSALITY IN RANDOM MATRIX THEORY1

∗ ∗ BY ALAN EDELMAN ,A.GUIONNET AND S. PÉCHÉ† Massachusetts Institute of Technology∗ and University of Paris Diderot (Paris 7)†

In order to have a better understanding of ﬁnite random matrices with non-Gaussian entries, we study the 1/N expansion of local eigenvalue statistics in both the bulk and at the hard edge of the spectrum of random matrices. This gives valuable information about the smallest singular value not seen in universality laws. In particular, we show the dependence on the fourth moment (or the kurtosis) of the entries. This work makes use of the so-called complex Gaussian divisible ensembles for both Wigner and sample covariance matrices.

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SUPER-REPLICATION WITH NONLINEAR TRANSACTION COSTS AND VOLATILITY UNCERTAINTY

∗ ∗ BY PETER BANK ,1,YAN DOLINSKY†,1,2 AND SELIM GÖKAY ,1 Technische Universität Berlin∗ and Hebrew University†

We study super-replication of contingent claims in an illiquid market with model uncertainty. Illiquidity is captured by nonlinear transaction costs in discrete time and model uncertainty arises as our only assumption on stock price returns is that they are in a range specified by fixed volatility bounds. We provide a dual characterization of super-replication prices as a supremum of penalized expectations for the contingent claim’s payoff. We also describe the scaling limit of this dual representation when the number of trading periods increases to infinity. Hence, this paper complements the results in [Finance Stoch. 17 (2013) 447–475] and [Ann. Appl. Probab. 5 (1995) 198–221] for the case of model uncertainty.

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THE SNAPPING OUT BROWNIAN MOTION

BY ANTOINE LEJAY1 Inria Nancy Grand-Est

We give a probabilistic representation of a one-dimensional diffusion equation where the solution is discontinuous at 0 with a jump proportional to its ﬂux. This kind of interface condition is usually seen as a semi-permeable barrier. For this, we use a process called here the snapping out Brownian motion, whose properties are studied. As this construction is motivated by applications, for example, in brain imaging or in chemistry, a simulation scheme is also provided.

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BACKWARD STOCHASTIC DIFFERENTIAL EQUATION DRIVEN BY A MARKED POINT PROCESS: AN ELEMENTARY APPROACH WITH AN APPLICATION TO OPTIMAL CONTROL

∗ ∗ BY FULVIA CONFORTOLA ,1,MARCO FUHRMAN ,1 AND JEAN JACOD† Politecnico di Milano∗ and Université Pierre et Marie Curie† We address a class of backward stochastic differential equations on a bounded interval, where the driving noise is a marked, or multivariate, point process. Assuming that the jump times are totally inaccessible and a technical condition holds (see Assumption (A) below), we prove existence and uniqueness results under Lipschitz conditions on the coefficients. Some counter- examples show that our assumptions are indeed needed. We use a novel approach that allows reduction to a (finite or infinite) system of deterministic differential equations, thus avoiding the use of martingale representation theorems and allowing potential use of standard numerical methods. Finally, we apply the main results to solve an optimal control problem for a marked point process, formulated in a classical way.

REFERENCES

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MSC2010 subject classifications. Primary 60H10; secondary 93E20. Key words and phrases. Backward stochastic differential equations, marked point processes, stochastic optimal control. [10] CRÉPEY,S.andMATOUSSI, A. (2008). Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison. Ann. Appl. Probab. 18 2041–2069. MR2462558 [11] EL KAROUI,N.andHUANG, S.-J. (1997). A general result of existence and uniqueness of backward stochastic differential equations. In Backward Stochastic Differential Equations (Paris, 1995–1996) (N. El Karoui and L. Mazliak, eds.). Pitman Res. Notes Math. Ser. 364 27–36. Longman, Harlow. MR1752673 [12] EL KAROUI,N.,PENG,S.andQUENEZ, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71. MR1434407 [13] JACOD, J. (1974/75). Multivariate point processes: Predictable projection, Radon–Nikodým derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235–253. MR0380978 [14] JACOD,J.andMÉMIN, J. (1981). Weak and strong solutions of stochastic differential equations: Existence and stability. In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980) (R. Williams, ed.). Lecture Notes in Math. 851 169–212. Springer, Berlin. MR0620991 [15] JEANBLANC,M.andRÉVEILLAC, A. (2014). A note on BSDEs with singular driver coefficients. In Arbitrage, Credit and Informational Risks. Peking University Series in Mathe- matics 5. World Scientific, Hackensack, NJ. [16] KHARROUBI,I.andLIM, T. (2012). Progressive enlargement of Filtrations and Backward SDEs with jumps. Preprint. [17] PARDOUX,É.andPENG, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61. MR1037747 [18] SHEN,L.andELLIOTT, R. J. (2011). Backward stochastic differential equations for a single jump process. Stoch. Anal. Appl. 29 654–673. MR2812521 [19] TANG,S.J.andLI, X. J. (1994). Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 1447–1475. MR1288257 [20] XIA, J. (2000). Backward stochastic differential equation with random measures. Acta Math. Appl. Sin. Engl. Ser. 16 225–234. MR1779016 The Annals of Applied Probability 2016, Vol. 26, No. 3, 1774–1806 DOI: 10.1214/15-AAP1133 © Institute of Mathematical Statistics, 2016

ENTROPIC RICCI CURVATURE BOUNDS FOR DISCRETE INTERACTING SYSTEMS1

BY MAX FATHI AND JAN MAAS Université Paris 6 and Institute of Science and Technology Austria

We develop a new and systematic method for proving entropic Ricci curvature lower bounds for Markov chains on discrete sets. Using different methods, such bounds have recently been obtained in several examples (e.g., 1-dimensional birth and death chains, product chains, Bernoulli–Laplace models, and random transposition models). However, a general method to obtain discrete Ricci bounds had been lacking. Our method covers all of the examples above. In addition, we obtain new Ricci curvature bounds for zero- range processes on the complete graph. The method is inspired by recent work of Caputo, Dai Pra and Posta on discrete functional inequalities.

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HACK’S LAW IN A DRAINAGE NETWORK MODEL: A BROWNIAN WEB APPROACH

BY RAHUL ROY,KUMARJIT SAHA AND ANISH SARKAR Indian Statistical Institute Hack [Studies of longitudinal stream proﬁles in Virginia and Maryland (1957). Report], while studying the drainage system in the Shenandoah valley and the adjacent mountains of Virginia, observed a power law relation l ∼ a0.6 between the length l of a stream from its source to a divide and the area a of the basin that collects the precipitation contributing to the stream as tributaries. We study the tributary structure of Howard’s drainage network model of headward growth and branching studied by Gangopadhyay, Roy and Sarkar [Ann. Appl. Probab. 14 (2004) 1242–1266]. We show that the exponent of Hack’s law is 2/3 for Howard’s model. Our study is based on a scaling of the process whereby the limit of the watershed area of a stream is area of a Brownian excursion process. To obtain this, we deﬁne a dual of the model and show that under diffusive scaling, both the original network and its dual converge jointly to the standard Brownian web and its dual.

REFERENCES

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MSC2010 subject classifications. 60D05. Key words and phrases. Brownian excursion, Brownian meander, Brownian web, Hack’s law. [11] GIACOMETTI,A.,MARTITAN,A.,RIGON,R.,RINALDO,A.andRODRIGUEZ-ITURBE,I. (1996). On Hack’s law. Water Resour. Res. 32 3367–3374. [12] GRAY, D. M. (1961). Interrelationships of watershed characteristics. J. Geophys. Res. 66 1215– 1233. [13] HACK, J. T. (1957). Studies of longitudinal stream profiles in Virginia and Maryland. Report, U.S. Geol. Surv. Prof. Pap. 294-B. [14] HORTON, R. T. (1945). Erosional development of streams and their drainage basins: Hydro- physical approach to quantitative morphology. Geol. Soc. Am. Bull. 56 275–370. [15] HURST, H. (1927). The Lake Plateau Basin of the Nile. Government Press, Cairo. [16] KAIGH, W. D. (1976). An invariance principle for random walk conditioned by a late return to zero. Ann. Probab. 4 115–121. MR0415706 [17] LANGBEIN, W. B. (1947). Topographic characteristics of drainage basins. Report, U.S. Geol. Surv. Prof. Pap. 968-C. [18] LEOPOLD,L.B.andLANGBEIN, W. B. (1962). The concept of entropy in landscape evolution. Report, U.S. Geol. Surv. Prof. Pap. 500-A. [19] LEVIN,D.A.,PERES,Y.andWILMER, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI. MR2466937 [20] LOUCHARD,G.andJANSON, S. (2007). Tail estimates for the Brownian excursion area and other Brownian areas. Electron. J. Probab. 12 1600–1632. MR2365879 [21] MANDELBROT, B. B. (1982). The Fractal Geometry of Nature.W.H.Freeman,NewYork. MR0665254 [22] MULLER, J. E. (1973). Re-evaluation of the relationship of master streams and drainage basins: Reply. Geol. Soc. Am. Bull. 84 3127–3130. [23] NEWMAN,C.M.,RAVISHANKAR,K.andSUN, R. (2005). Convergence of coalescing non- simple random walks to the Brownian web. Electron. J. Probab. 10 21–60. MR2120239 [24] REIMER, D. (2000). Proof of the van den Berg–Kesten conjecture. Combin. Probab. Comput. 9 27–32. MR1751301 [25] RESNICK, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modelling. Springer, New York. [26] RODRIGUEZ-ITURBE,I.andRINALDO, A. (1997). Fractal River Basins: Chance and Self- Organization. Cambridge Univ. Press, New York. [27] ROY,R.,SAHA,K.andSARKAR, A. (2015). Hack’s law in a drainage network model: A Brownian web approach. Preprint. Available at arXiv:1501.01382. [28] SOUCALIUC,F.,TÓTH,B.andWERNER, W. (2000). Reflection and coalescence between independent one-dimensional Brownian paths. Ann. Inst. Henri Poincaré Probab. Stat. 36 509–545. MR1785393 [29] SUN,R.andSWART, J. M. (2008). The Brownian net. Ann. Probab. 36 1153–1208. MR2408586 [30] TÓTH,B.andWERNER, W. (1998). The true self-repelling motion. Probab. Theory Related Fields 111 375–452. MR1640799 The Annals of Applied Probability 2016, Vol. 26, No. 3, 1837–1887 DOI: 10.1214/15-AAP1135 © Institute of Mathematical Statistics, 2016

GAUSSIAN FLUCTUATIONS FOR LINEAR SPECTRAL STATISTICS OF LARGE RANDOM COVARIANCE MATRICES

BY JAMAL NAJIM1 AND JIANFENG YAO2 Université Paris Est and CNRS and Télécom Paristech / Consider a N × n matrix = √1 R1 2X ,whereR is a nonnega- n n n n n tive deﬁnite Hermitian matrix and Xn is a random matrix with i.i.d. real or complex standardized entries. The ﬂuctuations of the linear statistics of the eigenvalues

N ∗ = ∗ Trace f nn f(λi), (λi) eigenvalues of nn, i=1

are shown to be Gaussian, in the regime where both dimensions of matrix n go to inﬁnity at the same pace and in the case where f is of class C3,thatis, has three continuous derivatives. The main improvements with respect to Bai and Silverstein’s CLT [Ann. Probab. 32 (2004) 553–605] are twofold: First, we consider general entries with ﬁnite fourth moment, but whose fourth cu- mulant is nonnull, that is, whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable. As a consequence, extra terms proportional to |V|2 = E n 22 = E n 4 −|V|2 − X11 and κ X11 2 appear in the limiting variance and in the limiting bias, which not only depend on the spectrum of matrix Rn but also on its eigenvectors. Second, we relax the analyticity assumption over f by representing the linear statistics with the help of Helffer–Sjöstrand’s formula. The CLT is expressed in terms of vanishing Lévy–Prohorov distance between the linear statistics’ distribution and a Gaussian probability distribution, the mean and the variance of which depend upon N and n and may not converge.

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DUALITY THEORY FOR PORTFOLIO OPTIMISATION UNDER TRANSACTION COSTS

BY CHRISTOPH CZICHOWSKY1 AND WALTER SCHACHERMAYER2 London School of Economics and Political Science and Universität Wien

We consider the problem of portfolio optimisation with general càdlàg price processes in the presence of proportional transaction costs. In this con- text, we develop a general duality theory. In particular, we prove the existence of a dual optimiser as well as a shadow price process in an appropriate generalised sense. This shadow price is deﬁned by means of a “sandwiched” process consisting of a predictable and an optional strong supermartingale, and pertains to all strategies that remain solvent under transaction costs. We provide examples showing that, in the general setting we study, the shadow price processes have to be of such a generalised form.

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MR1883202 The Annals of Applied Probability 2016, Vol. 26, No. 3, 1942–1946 DOI: 10.1214/15-AAP1121 © Institute of Mathematical Statistics, 2016

A NOTE ON THE EXPANSION OF THE SMALLEST EIGENVALUE DISTRIBUTION OF THE LUE AT THE HARD EDGE1

BY FOLKMAR BORNEMANN Technische Universität München In a recent paper, Edelman, Guionnet and Péché conjectured a particular − n 1 correction term of the smallest eigenvalue distribution of the Laguerre unitary ensemble (LUE) of order n in the hard-edge scaling limit: speciﬁ- cally, the derivative of the limit distribution, that is, the density, shows up in that correction term. We give a short proof by modifying the hard-edge − scaling to achieve an optimal O(n 2) rate of convergence of the smallest eigenvalue distribution. The appearance of the derivative follows then by a Taylor expansion of the less optimal, standard hard-edge scaling. We relate − the n 1 correction term further to the logarithmic derivative of the Bessel kernel Fredholm determinant in the work of Tracy and Widom.

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